The Diophantine equation for which p is prime 2^n = p^2 + 7 and n is integers with p > 0, is examined in this work. It is demonstrated by examining the characteristics of the equation that the only solutions meeting these requirements are p = 3 with n = 5 and p = 5 with n = 7. In order to analyze the above equation, it is necessary to look at the properties of primes and how they relate to powers of two. These results advance our knowledge of prime numbers and how they behave in certain mathematical problems.

In this paper, we examine the weak abc conjecture and the strong abc conjecture using some functions. we attempt to discuss the possibilty that the negative consequence of the weak abc conjecture. Namely, we derive that the weak abc conjecture is not true. Additionally, we show that the strong abc conjecture is not true.

When discussing the Diophantine quintic equation $p(a^5 + b^5) = q(c^5 + d^5)$ where p is a prime and q is an integer, there is a clear gap between the mathematical literature and online forums. Because of the intrinsic complexity of parameterizing fifth-degree equations, this equation re- mains largely unexplored. In this work, we approach this quintic problem through algebraic methods with the goal of providing numerical solutions that illuminate its properties and behaviors.Our research uncovers in- triguing patterns and connections that shed light on the enigmatic nature of Diophantine quintic equations in this particular form.

Deep learning machines are computational models composed of multiple processing layers of adaptive weights to learn representations of data with multiple levels of abstraction. Their structure is mainly reflecting the intuitive plausibility of decomposing a problem into multiple levels of computation and representation since it is believed that higher layers of representation allow a system to learn complex functions. Surprisingly, after decades of research, from learning and design perspectives these models are still deployed in a heuristic manner. In this paper, deep learning feed-forward machines are modeled from a statistical mechanics point of view as disordered physical systems where its macroscopic behavior is determined in terms of the interactions defined between the basic constituent of these models, namely, the artificial neuron. They are viewed as the equilibrium states of a theoretical body that is subject to the law of increase of the entropy. The study of the changes in energy of the body when passing from one equilibrium state to another is used to understand the structure and role of the phase space of the system, the stability of the equilibrium states, and the resulting degree of disorder. It is shown that the topology of these models is strongly linked to their stability and resulting level of disorder. Furthermore, the proposed theoretical characterization permit to assess the thermodynamic efficiency with which information can be processed by these models, and to provide a practical methodology to quantitatively estimate and compare their expected learning and generalization capabilities. These theoretical results provides new insights to the theory of deep learning and their implications are shown to be consistent through a set of benchmarks designed to experimentally assess their validity.

For any positive integer n ≥ 2, the Erdős-Straus Conjecture claims that the Diophantine equation 4/n = 1/x + 1/y + 1/z has a solution where x, y, z are also positive integers. In this paper, a directed network based on this equation is generated, with properties such as its average clustering coefficient, average path length, degree distributions, and largest strongly connected component analysed to reveal some underlying trends about the nature of the conjecture. Potential connections between different numbers, that result from satisfying a source-solution relationship for this equation, are described using the appropriate number-theoretic interpretations wherever possible, while conjectures backed by these trends are made in other instances. Additionally, a directed configuration model is used to show that the origin of several results is the degree sequence of the network. Metrics relating to the prime number nodes, specifically their in and out degrees, are also explored to yield some intriguing observations. On the whole, the aim is to highlight the viability of complex networks as a computational tool to study this general class of problems pertaining to fixed-length unit fraction splits.

The paper considers two models of queuing with a varying structure based on the introduction of additional transient intensities into known models or their combinations, which create stationary distributions convenient for calculation. In the first model, it is a probabilistic mixture of known stationary distributions with given weights. In the second model, this uniform distribution is repeatedly used in physical statistics. Both models are based on the selection of states, between which additional transient intensities are introduced. The algorithms used in the paper for introducing new transient intensities are closely related to the concept of flow in a deterministic transport network. The introduced controls are selected so that the marginal distribution of the combined system is a mixture of the marginal distributions of the combined systems with different weights determined by the introduced transient intensities. As a result, the process of functioning of the combined system is obtained by switching processes corresponding to different combined systems at certain points in time.

In this paper defines the consecutive sum of the digits of a natural number, so far as it becomes less than ten, as an arithmetic function called and then introduces some important properties of this function by proving a few theorems in a way that they can be used as a powerful tool in many cases. As an instance, by introducing a test called test, it has been shown that we are able to examine many algebraic equalities in the form of in which and are arithmetic functions and to easily study many of the algebraic and diophantine equations in the domain of whole numbers. The importance of test for algebraic equalities can be considered equivalent to dimensional equation in physics relations and formulas. Additionally, this arithmetic function can also be useful in factorizing the composite odd numbers.

Let a,b,c be positive integers such that a²+b²=c², 2|b,gcd(a,b)=1. In 1956, Jesmanowicz conjectured that for any positive integer w, the only solution of (aw)^x+(bw)^y=(cw)^z in positive integers is (x, y, z) = (2, 2, 2). In this paper, based on Gaussian integer ring, we show that Je$\acute{s}$manowicz' conjecture is true for any positive integer

We prove: (1) the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable, (2) the set of all Diophantine equations which have at most finitely many solutions in positive integers is not recursively enumerable, (3) the set of all Diophantine equations which have at most finitely many integer solutions is not recursively enumerable, (4) analogous theorems hold for Diophantine equations D(x₁, …, x_p) = 0, where p ∈ N\{0} and for every i ∈ {1, …, p} the polynomial D(x₁, …, x_p) involves a monomial M with a non-zero coefficient such that x_i divides M, (5) the set of all Diophantine equations which have at most k variables (where k ≥ 9) and at most finitely many solutions in non-negative integers is not recursively enumerable.

We suggest progress regarding the following six physics opportunities. List all elementary particles. Describe dark matter. Explain ratios of dark matter to ordinary matter. Explain eras in the history of the universe. Link properties of objects. Interrelate physics models. We use models based on Diophantine equations.

Let $f\left(1\right)=1$ , and let $f(n+1)=2^{2^{f\left(n\right)}}$ for every positive integer n. We consider the following hypothesis: if a system S ⊆ {x_i · x_j = x_k : i, j, k ∈ {1, . . . , n}} ∪ {x_i + 1 = x_k : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x₁, . . . , x_n, then each such solution (x₁, . . . , x_n) satisfies x₁, . . . , x_n ≤ f (2n). We prove: (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set $\mathfrak{M}$ ⊆ has a finite-fold Diophantine representation, then $\mathfrak{M}$ is computable.

Sums of M consecutive squared integers (a+i)^2 equaling squared integers (for a<=1, 0<= i<= M-1) yield remarkable regular linear features when plotting values of M in function of a. These features correspond to groupings of pairs of a values for successive same values of M around straight lines of equation mu*M = 2a and are characterized in this paper for rational values of mu.

The notion of potential-growth indicator came to being in the field of matrix population models long ago, almost simultaneously with the pioneering Leslie model for age-structured population dynamics, albeit the term has been given and the theory developed only recent years. The indicator represents an explicit function, R(L), of matrix L elements and indicates the position of the spectral radius of L relative to 1 on the real axis, thus signifying the population growth, or decline, or stabilization. Some indicators turned out useful in theoretical layouts and practical applications prior to calculating the spectral radius itself. The most senior (1994) and popular indicator, R0(L), is known as the net reproductive rate, and we consider two more ones, R1(L) and RRT(A), developed later on. All the three are different in what concerns their simplicity and the level of generality, and we illustrate them with a case study of Calamagrostis epigeios, a long-rhizome perennial weed actively colonizing open spaces in the temperate zone. While the R0(L) and R1(L) fail respectively because of complexity and insufficient generality, the RRT(L) does succeed, justifying the merit of indication.

Matiyasevich's theorem states that there is no algorithm to decide whether or not a given Diophantine equation has a solution in non-negative integers. Smorynski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. We prove: (1) Smorynski's theorem easily follows from Matiyasevich's theorem, (2 ) Hilbert's Tenth Problem for Q has a negative solution if and only if the set of all Diophantine equations with a finite number of rational solutions is not recursively enumerable.